Abc conjecture

French mathematician Joseph Oesterlé

British mathematician David Masser

The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985). It is stated in terms of three positive integers, a, b and c (hence the name) that are relatively prime and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is usually not much smaller than c. In other words: if a and b are composed from large powers of primes, then c is usually not divisible by large powers of primes. The precise statement is given below.

The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves. The latter conjecture has more geometric structures involved in its statement in comparison with the abc conjecture

The abc conjecture and its versions express, in concentrate form, some key feature of various problems in Diophantine geometry. A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis".

Lucien Szpiro attempted a solution in 2007, but it was found to be incorrect.

In August 2012 Shinichi Mochizuki posted his four preprints which develop a new inter-universal Teichmüller theory (IUT) based on anabelian geometry. The fourth paper applied IUT to obtain the proof of several famous conjectures including the abc conjecture. Mochizuki's papers were submitted to a mathematical journal and are being refereed. Substantial efforts have been undertaken to help other mathematicians to study IUT, and two international conferences were held.

The number of experts on IUT keeps growing.

Formulations

Before we state the conjecture we need to introduce the notion of the radical of an integer: for a positive integer n, the radical of n, denoted rad(n), is the product of the distinct prime factors of n. For example

rad(16) = rad(2⁴) = 2,

rad(17) = 17,

rad(18) = rad(2 ⋅ 3²) = 2 · 3 = 6,

rad(1000000) = rad(2⁶ ⋅ 5⁶) = 2 ⋅ 5 = 10.

If a, b, and c are coprime positive integers such that a + b = c, it turns out that "usually" c < rad(abc). The abc conjecture deals with the exceptions. Specifically, it states that:

ABC Conjecture. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that:

$c>\operatorname {rad} (abc)^{1+\varepsilon }.$

An equivalent formulation states that:

ABC Conjecture II. For every ε > 0, there exists a constant K_ε such that for all triples (a, b, c) of coprime positive integers, with a + b = c:

$c<K_{\varepsilon }\cdot \operatorname {rad} (abc)^{1+\varepsilon }.$

A third equivalent formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), defined as

$q(a,b,c)={\frac {\log(c)}{\log(\operatorname {rad} (abc))}}.$

For example,

q(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...

q(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...

A typical triple (a, b, c) of coprime positive integers with a + b = c will have c < rad(abc), i.e. q(a, b, c) < 1. Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers.

ABC Conjecture III. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε.

Whereas it is known that there are infinitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1, the conjecture predicts that only finitely many of those have q > 1.01 or q > 1.001 or even q > 1.0001, etc. In particular, if the conjecture is true then there must exist a triple (a, b, c) which achieves the maximal possible quality q(a, b, c) .

Examples of triples with small radical

The condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with rad(abc) < c. For example let:

$a=1,\quad b=2^{6n}-1,\quad c=2^{6n},\qquad n>1.$

First we note that b is divisible by 9:

$b=2^{6n}-1=64^{n}-1=(64-1)(\cdots )=9\cdot 7\cdot (\cdots )$

Using this fact we calculate:

${\begin{aligned}\operatorname {rad} (abc)&=\operatorname {rad} (a)\operatorname {rad} (b)\operatorname {rad} (c)\\&=\operatorname {rad} (1)\operatorname {rad} \left(2^{6n}-1\right)\operatorname {rad} \left(2^{6n}\right)\\&=2\operatorname {rad} \left(2^{6n}-1\right)\\&=2\operatorname {rad} \left(9\cdot {\tfrac {b}{9}}\right)\\&\leqslant 2\cdot 3\cdot {\tfrac {b}{9}}\\&=2{\tfrac {b}{3}}\\&<{\tfrac {2}{3}}c\end{aligned}}$

By replacing the exponent 6n by other exponents forcing b to have larger square factors, the ratio between the radical and c can be made arbitrarily small. Specifically, let p > 2 be a prime and consider:

$a=1,\quad b=2^{p(p-1)n}-1,\quad c=2^{p(p-1)n},\qquad n>1.$

Now we claim that b is divisible by p²:

${\begin{aligned}b&=2^{p(p-1)n}-1\\&=\left(2^{p(p-1)}\right)^{n}-1\\&=\left(2^{p(p-1)}-1\right)(\cdots )\\&=p^{2}\cdot r(\cdots )\end{aligned}}$

The last step uses the fact that p² divides 2^p(p-1)-1. This follows from Fermat's little theorem, which shows that, for p>2, 2^p-1=pk+1 for some integer k. Raising both sides to the power of p then shows that 2^p(p-1)=p²(...)+1.

And now with a similar calculation as above we have:

$\operatorname {rad} (abc)<{\tfrac {2}{p}}c$

A list of the highest-quality triples (triples with a particularly small radical relative to c) is given below; the highest quality, 1.6299, was found by Eric Reyssat (Lando & Zvonkin 2004, p. 137) for

a = 2,

b = 3¹⁰·109 = 6,436,341,

c = 23⁵ = 6,436,343,

rad(abc) = 15042.

Some consequences

The abc conjecture has a large number of consequences. These include both known results (some of which have been proven separately since the conjecture has been stated) and conjectures for which it gives a conditional proof. While an earlier proof of the conjecture would have been more significant in terms of consequences, the abc conjecture itself remains of interest for the other conjectures it would prove, together with its numerous links with deep questions in number theory.

• Thue–Siegel–Roth theorem on diophantine approximation of algebraic numbers (Bombieri 1994)
• The Mordell conjecture (already proven in general by Gerd Faltings) (Elkies 1991)
• It is equivalent to Vojta's conjecture (in dimension 1). (Van Frankenhuijsen 2002)
• The Erdős–Woods conjecture except for a finite number of counterexamples (Langevin 1993)
• The existence of infinitely many non-Wieferich primes in every base b > 1 (Silverman 1988)
• The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers (Nitaj 1996)
• The Fermat–Catalan conjecture, a generalization of Fermat's last theorem concerning powers that are sums of powers (Pomerance 2008)
• The L-function L(s, χ_d) formed with the Legendre symbol, has no Siegel zero (this consequence actually requires a uniform version of the abc conjecture in number fields, not only the abc conjecture as formulated above for rational integers) (Granville & Stark 2000)
• P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros.
• A generalization of Tijdeman's theorem concerning the number of solutions of y^m = xⁿ + k (Tijdeman's theorem answers the case k = 1), and Pillai's conjecture (1931) concerning the number of solutions of Ay^m = Bxⁿ + k.
• It is equivalent to the Granville–Langevin conjecture, that if f is a square-free binary form of degree n > 2, then for every real β > 2 there is a constant C(f, β) such that for all coprime integers x, y, the radical of f(x, y) exceeds C · max{|x|, |y|}^n−β.
• It is equivalent to the modified Szpiro conjecture, which would yield a bound of rad(abc)^1.2+ε (Oesterlé 1988).
• Dąbrowski (1996) has shown that the abc conjecture implies that the Diophantine equation n! + A = k² has only finitely many solutions for any given integer A.
• There are ~c_fN positive integers n ≤ N for which f(n)/B' is square-free, with c_f > 0 a positive constant defined as: (Granville 1998)

$c_{f}=\prod _{{\text{prime }}p}x_{i}\left(1-{\frac {\omega \,\!_{f}(p)}{p^{2+q_{p}}}}\right).$

• Fermat's Last Theorem has a famously difficult proof by Andrew Wiles. However it follows easily, at least for $n\geq 6$, from an effective form of a weak version of the abc conjecture. The abc conjecture says the lim sup of the set of all qualities (defined above) is 1, which implies the much weaker assertion that there is a finite upper bound for qualities. The conjecture that 2 is such an upper bound suffices for a very short proof of Fermat's Last Theorem for $n\geq 6$.
• The Beal conjecture, a generalization of Fermat's last theorem proposing that if A, B, C, x, y, and z are positive integers with A^x + B^y = C^z and x, y, z > 2, then A, B, and C have a common prime factor.

Theoretical results

The abc conjecture implies that c can be bounded above by a near-linear function of the radical of abc. However, exponential bounds are known. Specifically, the following bounds have been proven:

$c<\exp {\left(K_{1}\operatorname {rad} (abc)^{15}\right)}$ (Stewart & Tijdeman 1986),

$c<\exp {\left(K_{2}\operatorname {rad} (abc)^{{\frac {2}{3}}+\varepsilon }\right)}$ (Stewart & Yu 1991), and

$c<\exp {\left(K_{3}\operatorname {rad} (abc)^{{\frac {1}{3}}+\varepsilon }\right)}$ (Stewart & Yu 2001).

In these bounds, K₁ is a constant that does not depend on a, b, or c, and K₂ and K₃ are constants that depend on ε (in an effectively computable way) but not on a, b, or c. The bounds apply to any triple for which c > 2.

Computational results

In 2006, the Mathematics Department of Leiden University in the Netherlands, together with the Dutch Kennislink science institute, launched the ABC@Home project, a grid computing system, which aims to discover additional triples a, b, c with rad(abc) < c. Although no finite set of examples or counterexamples can resolve the abc conjecture, it is hoped that patterns in the triples discovered by this project will lead to insights about the conjecture and about number theory more generally.

Distribution of triples with q > 1

	q > 1	q > 1.05	q > 1.1	q > 1.2	q > 1.3	q > 1.4
c < 102	6	4	4	2	0	0
c < 103	31	17	14	8	3	1
c < 104	120	74	50	22	8	3
c < 105	418	240	152	51	13	6
c < 106	1,268	667	379	102	29	11
c < 107	3,499	1,669	856	210	60	17
c < 108	8,987	3,869	1,801	384	98	25
c < 109	22,316	8,742	3,693	706	144	34
c < 1010	51,677	18,233	7,035	1,159	218	51
c < 1011	116,978	37,612	13,266	1,947	327	64
c < 1012	252,856	73,714	23,773	3,028	455	74
c < 1013	528,275	139,762	41,438	4,519	599	84
c < 1014	1,075,319	258,168	70,047	6,665	769	98
c < 1015	2,131,671	463,446	115,041	9,497	998	112
c < 1016	4,119,410	812,499	184,727	13,118	1,232	126
c < 1017	7,801,334	1,396,909	290,965	17,890	1,530	143
c < 1018	14,482,065	2,352,105	449,194	24,013	1,843	160

ABC@Home had found 23.8 million triples.

Highest quality triples

	q	a	b	c	Discovered by
1	1.6299	2	310·109	235	Eric Reyssat
2	1.6260	112	32·56·73	221·23	Benne de Weger
3	1.6235	19·1307	7·292·318	28·322·54	Jerzy Browkin, Juliusz Brzezinski
4	1.5808	283	511·132	28·38·173	Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj
5	1.5679	1	2·37	54·7	Benne de Weger

Note: the quality q(a, b, c) of the triple (a, b, c) is defined above.

Refined forms, generalizations and related statements

The abc conjecture is an integer analogue of the Mason–Stothers theorem for polynomials.

A strengthening, proposed by Baker (1998), states that in the abc conjecture one can replace rad(abc) by

ε^−ω rad(abc),

where ω is the total number of distinct primes dividing a, b and c (Bombieri & Gubler 2006, p. 404).

Andrew Granville noticed that the minimum of the function $({\varepsilon }^{-\omega }\operatorname {rad} (abc))^{1+\varepsilon }$ over ${\varepsilon }>0$ occurs when ${\varepsilon }={\frac {\omega }{\log(\operatorname {rad} (abc))}}.$

This incited Baker (2004) to propose a sharper form of the abc conjecture, namely:

$c<{\kappa }\operatorname {rad} (abc){\frac {(\log(\operatorname {rad} (abc)))^{\omega }}{\omega !}}$

with κ an absolute constant. After some computational experiments he found that a value of ${\tfrac {6}{5}}$ was admissible for κ.

This version is called "explicit abc conjecture".

From the previous inequality, Baker deduced a stronger form of the original abc conjecture: let a, b, c be coprime positive integers with a + b = c; then we have:

$c<(\operatorname {rad} (abc))^{1+{\frac {3}{4}}}$.

Baker (1998) also describes related conjectures of Andrew Granville that would give upper bounds on c of the form

$K^{\Omega (abc)}\mathrm {rad} (abc),$

where Ω(n) is the total number of prime factors of n and

$O(\mathrm {rad} (abc)\Theta (abc)),$

where Θ(n) is the number of integers up to n divisible only by primes dividing n.

Robert, Stewart & Tenenbaum (2014) proposed more precise inequality based on Robert & Tenenbaum (2013). Let k = rad(abc). They conjectured there is a constant C₁ such that

$c<k\exp \left(4{\sqrt {\frac {3\log k}{\log \log k}}}\left(1+{\frac {\log \log \log k}{2\log \log k}}+{\frac {C_{1}}{\log \log k}}\right)\right)$

holds whereas there is a constant C₂ such that

$c>k\exp \left(4{\sqrt {\frac {3\log k}{\log \log k}}}\left(1+{\frac {\log \log \log k}{2\log \log k}}+{\frac {C_{2}}{\log \log k}}\right)\right)$

holds infinitely often.

Browkin & Brzeziński (1994) formulated the n conjecture—a version of the abc conjecture involving n > 2 integers.

The work of Shinichi Mochizuki

In August 2012, Shinichi Mochizuki released a series of four preprints on Inter-universal Teichmuller Theory which is then applied to prove several famous conjectures in number theory, including Szpiro's conjecture, the hyperbolic Vojta's conjecture and the abc conjecture. Mochizuki calls the theory on which this proof is based "inter-universal Teichmüller theory (IUT)". The theory is radically different from any standard theories and goes well outside the scope of arithmetic geometry. It was developed over two decades with the last four IUT papers occupying the space of over 500 pages and using many of his prior published papers.

Mochizuki released progress reports in December 2013 and December 2014. He has invested hundreds of hours to run seminars and meetings to discuss his theory. According to Mochizuki, verification of the core proof is "for all practical purposes, complete." However, he also stated that an official declaration should not happen until some time later in the 2010s, due to the importance of the results and new techniques.

The first international workshop on Mochizuki's theory was organized by Ivan Fesenko and held in Oxford in December 2015. It helped to increase the number of mathematicians who had thoroughly studied parts of the IUT papers or related prerequisite papers. The next workshop on IUT Summit was held at the Research Institute for Mathematical Sciences in Kyoto in July 2016. After that workshop at least ten mathematicians now understand the theory in detail. There are several introductory texts and surveys of the theory, written by Mochizuki and other mathematicians.

See also

• List of unsolved problems in mathematics

References

• Baker, Alan (1998). "Logarithmic forms and the abc-conjecture". In Győry, Kálmán (ed.). Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996. Berlin: de Gruyter. pp. 37–44. ISBN 3-11-015364-5. Zbl 0973.11047. {{cite book}}: Invalid |ref=harv (help)
• Baker, Alan (2004). "Experiments on the abc-conjecture". Publ. Math. Debrecen. 65: 253–260. {{cite journal}}: Invalid |ref=harv (help)
• Bombieri, Enrico (1994). "Roth's theorem and the abc-conjecture". preprint. ETH Zürich. {{cite journal}}: Invalid |ref=harv (help)
• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034. {{cite book}}: Invalid |ref=harv (help)
• Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. doi:10.2307/2153551. JSTOR 2153551. {{cite journal}}: Invalid |ref=harv (help)
• Browkin, Jerzy (2000). "The abc-conjecture". In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. (eds.). Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 75–106. ISBN 3-7643-6259-6. {{cite book}}: Invalid |ref=harv (help)
• Dąbrowski, Andrzej (1996). "On the diophantine equation x! + A = y²". Nieuw Archief voor Wiskunde, IV. 14: 321–324. Zbl 0876.11015. {{cite journal}}: Invalid |ref=harv (help)
• Elkies, N. D. (1991). "ABC implies Mordell". Intern. Math. Research Notices. 7 (7): 99–109. doi:10.1155/S1073792891000144. {{cite journal}}: Invalid |ref=harv (help)CS1 maint: unflagged free DOI (link)
• Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons (September): 26–34. JSTOR 25678079. {{cite journal}}: Invalid |ref=harv (help)
• Goldfeld, Dorian (2002). "Modular forms, elliptic curves and the abc-conjecture". In Wüstholz, Gisbert (ed.). A panorama in number theory or The view from Baker's garden. Based on a conference in honor of Alan Baker's 60th birthday, Zürich, Switzerland, 1999. Cambridge: Cambridge University Press. pp. 128–147. ISBN 0-521-80799-9. Zbl 1046.11035. {{cite book}}: Invalid |ref=harv (help)
• Gowers, Timothy; Barrow-Green, June; Leader, Imre, eds. (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. pp. 361–362, 681. ISBN 978-0-691-11880-2. {{cite book}}: Invalid |ref=harv (help)
• Granville, A. (1998). "ABC Allows Us to Count Squarefrees" (PDF). International Mathematics Research Notices. 1998: 991–1009. doi:10.1155/S1073792898000592. {{cite journal}}: Invalid |ref=harv (help)CS1 maint: unflagged free DOI (link)
• Granville, Andrew; Stark, H. (2000). "ABC implies no "Siegel zeros" for L-functions of characters with negative exponent" (PDF). Inventiones Mathematicae. 139: 509–523. doi:10.1007/s002229900036. {{cite journal}}: Invalid |ref=harv (help)
• Granville, Andrew; Tucker, Thomas (2002). "It's As Easy As abc" (PDF). Notices of the AMS. 49 (10): 1224–1231. {{cite journal}}: Invalid |ref=harv (help)
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Berlin: Springer-Verlag. ISBN 0-387-20860-7. {{cite book}}: Invalid |ref=harv (help)
• Lando, Sergei K.; Zvonkin, Alexander K. (2004). Graphs on Surfaces and Their Applications. Vol. 141. Springer-Verlag. ISBN 3-540-00203-0. {{cite book}}: |work= ignored (help); Invalid |ref=harv (help)
• Langevin, M. (1993). "Cas d'égalité pour le théorème de Mason et applications de la conjecture abc". Comptes rendus de l'Académie des sciences (in French). 317 (5): 441–444. {{cite journal}}: Invalid |ref=harv (help)
• Masser, D. W. (1985). "Open problems". In Chen, W. W. L. (ed.). Proceedings of the Symposium on Analytic Number Theory. London: Imperial College. {{cite book}}: Invalid |ref=harv (help)
• Mollin, R.A. (2009). "A note on the ABC-conjecture" (PDF). Far East J. Math. Sci. 33 (3): 267–275. ISSN 0972-0871. Zbl 1241.11034.
• Mollin, Richard A. (2010). Advanced number theory with applications. Boca Raton, FL: CRC Press. ISBN 978-1-4200-8328-6. Zbl 1200.11002.
• Nitaj, Abderrahmane (1996). "La conjecture abc". Enseign. Math. (in French). 42 (1–2): 3–24. {{cite journal}}: Invalid |ref=harv (help)
• Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat", Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179, MR 0992208
• Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362. {{cite book}}: Invalid |ref=harv (help)
• Silverman, Joseph H. (1988). "Wieferich's criterion and the abc-conjecture". Journal of Number Theory. 30 (2): 226–237. doi:10.1016/0022-314X(88)90019-4. Zbl 0654.10019. {{cite journal}}: Invalid |ref=harv (help)
• Robert, Olivier; Stewart, Cameron L.; Tenenbaum, Gérald (2014). "A refinement of the abc conjecture". Bull. London Math. Soc. 46 (6): 1156–1166. doi:10.1112/blms/bdu069. {{cite journal}}: Invalid |ref=harv (help)
• Robert, Olivier; Tenenbaum, Gérald (2013). "Sur la répartition du noyau d'un entier". Indag. Math. 24: 802–914. doi:10.1016/j.indag.2013.07.007. {{cite journal}}: Invalid |ref=harv (help)
• Stewart, C. L.; Tijdeman, R. (1986). "On the Oesterlé-Masser conjecture". Monatshefte für Mathematik. 102 (3): 251–257. doi:10.1007/BF01294603. {{cite journal}}: Invalid |ref=harv (help)
• Stewart, C. L.; Yu, Kunrui (1991). "On the abc conjecture". Mathematische Annalen. 291 (1): 225–230. doi:10.1007/BF01445201. {{cite journal}}: Invalid |ref=harv (help)
• Stewart, C. L.; Yu, Kunrui (2001). "On the abc conjecture, II". Duke Mathematical Journal. 108 (1): 169–181. doi:10.1215/S0012-7094-01-10815-6. {{cite journal}}: Invalid |ref=harv (help)
• Van Frankenhuijsen, Machiel (2002). "The ABC conjecture implies Vojta's height inequality for curves". J. Number Theory. 95 (2): 289–302. doi:10.1006/jnth.2001.2769. MR 1924103. {{cite journal}}: Invalid |ref=harv (help)

External links

• ABC@home Distributed computing project called ABC@Home.
• Easy as ABC: Easy to follow, detailed explanation by Brian Hayes.
• Weisstein, Eric W. "abc Conjecture". MathWorld.
• Abderrahmane Nitaj's ABC conjecture home page
• Bart de Smit's ABC Triples webpage
• http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
• The ABC's of Number Theory by Noam D. Elkies
• Questions about Number by Barry Mazur
• Philosophy behind Mochizuki’s work on the ABC conjecture on MathOverflow
• ABC Conjecture Polymath project wiki page linking to various sources of commentary on Mochizuki's papers.
• abc Conjecture Numberphile video
• News about IUT by Mochizuki